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8). , and L denote lengths in the initial and subsequent states, respectively. The deformation of a medium is determined by the displacements of particles and lines of particles. 2. After deformation, the particle is at P ∗ with the position vector R = R(θ1 , θ2 , θ3 ; t) = Xi (θ1 , θ2 , θ3 ; t)ˆıi . 2) asserts that the terminal position of the particle depends on its initial coordinates and time. The continuity of the medium requires that R be continuous. We assume that its derivatives are also continuous.

51b) serve to deﬁne the covariant derivatives of the contravariant (V i ) and covariant (Vi ) components of a vector. Observe that the covariant derivative (V j |i or Vj |i ) plays the same role as the partial derivative (Vj,i ) plays in the Cartesian coordinate system, that the base vector (g i or g i ) plays the same role as the unit vector ˆıi in the Cartesian system, and that the metric tensor (gij or g ij ) reduces to the Kronecker delta δij in the Cartesian coordinates. 46a), the Christoﬀel symbol Γijk in one coordinate system θ¯i is expressed in terms of the symbols in another system θi by the formula: Γijk = Γlmn ∂θl ∂θm ∂θn ∂θl ∂ 2 θm + glm .